That is heartbreaking! I am so very sorry, the poor thing.
I have a suggestion: buy a dropper bottle of GRAPEFRUIT 'SEED' EXTRACT.
Do NOT ever use it straight nor get in the eyes; put a couple of drops in her drink twice a day. Then mix a few drops in a bowl of warm water. Gently pour it over her eczema. You could also buy a small spray bottle at walmart and mix distilled water that you have warmed and added a few drops of Grapefruit-Seed-Extract to and spray it on. If Grapefruit-Seed-Extract is not used by itself it is NOT damaging or dangerous. In fact, it prevents and cures many ailments. Hope this helps.
Remember: think in terms of DROPS only.